Boolean Circuits Of Polynomial Size Research Articles

Efficient deterministic MapReduce algorithms for parallelizable problems

The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be notoriously difficult, however. In continuation of Goodrich et al.'s early efforts, explicitly espousing the goal of putting the MapReduce framework on footing equal to that of long-established models such as the PRAM, we investigate the obvious complexity question of how the computational power of MapReduce algorithms compares to that of combinational Boolean circuits commonly used for parallel computations. Relying on the standard MapReduce model introduced by Karloff et al. a decade ago, we develop an intricate simulation technique to show that any problem in NC (i.e., a problem solved by a logspace-uniform family of Boolean circuits of polynomial size and a depth polylogarithmic in the input size) can be solved by a MapReduce computation in O(T(n)/log⁡n) rounds, where n is the input size and T(n) is the depth of the witnessing circuit family. Thus, we are able to closely relate the standard, uniform NC hierarchy modeling parallel computations to the deterministic MapReduce hierarchy DMRC by proving that NCi+1⊆DMRCi for all i∈N. Besides the theoretical significance, this result has important applied aspects as well. In particular, we show for all problems in NC1—many practically relevant ones, such as integer multiplication and division, the parity function, and recognizing balanced strings of parentheses being among these—how to solve them in a constant number of deterministic MapReduce rounds.

A generalization of Spira's theorem and circuits with small segregators or separators

Spira showed that any Boolean formula of size s can be simulated in depth O(log⁡s). We generalize Spira's theorem and show that any Boolean circuit of size s with segregators (or separators) of size f(s) can be simulated in depth O(f(s)log⁡s). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O(k2log⁡n) by Jansen and Sarma. Our results imply that the class of languages computed by non-uniform families of polynomial-size circuits with constant size segregators equals non-uniform NC1.As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators is in deterministic SPACE(log2⁡n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete, is in SPACE(nlog⁡n); and that the Layered Circuit Value and Synchronous Circuit Value problems, which are both P-complete, are in SPACE(n).

The number of Boolean functions computed by formulas of a given size

Estimates are given of the number B(n, L) of distinct functions computed by propositional formulas of size L in n variables, constructed using only literals and ∧, ∨ connectives. (L is the number of occurrences of variables. L−1 is the number of binary ∧s and ∨s. B(n, L) is also the number of functions computed by two terminal series-parallel networks with L switches.) Enroute the read-once functions, which are closely related to Schroder numbers, are enumerated. Writing B(n, L)=b(n, L)L, we find that if L and β(n) go to infinity with increasing n and L≤2n/nβ(n), then b(n, L)∼cn, where c=2/(ln 4−1). Making a comparison with polynomial size Boolean circuits, this implies the following. For any constant α>1, almost all Boolean functions with formula complexity at most nα cannot be computed by any circuit constructed from literals and fewer than α−1nα two-input ∧, ∨ gates. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 349–382, 1998

On the Circuit Complexity of Sigmoid Feedforward Neural Networks

This paper aims to examine the circuit complexity of sigmoid activation feedforward artificial neural networks by placing them amongst several classic Boolean and threshold gate circuit complexity classes. The starting point is the class NNk defined by Shawe-Taylor et al. (1992) Classes of feedforward neural nets and their circuit complexity. Neural Networks 5(6), 971–977. For a better characterisation, we introduce two additional classes NNkΔ and NNkΔ,ϵ having less restrictive conditions than NNk concerning fan-in and accuracy, and proceed to prove relations amongst these three classes and well established circuit complexity classes. For doing that, a particular class of Boolean functions FΔ is first introduced and we show how a threshold gate circuit can be recursively built for any fΔ belonging to FΔ. As the G-functions (computing the carries) are fΔ functions, a class of solutions is obtained for threshold gate adders. We then constructively prove the inclusions amongst circuit complexity classes. This is done by converting the sigmoid feedforward artificial neural network into an equivalent threshold gate circuit Shawe-Taylor et al. (1992). Each threshold gate is then replaced by a multiple input adder having a binary tree structure, relaxing the logarithmic fan-in condition from (Shawe-Taylor et al. 1992) to (almost) polynomial. This means that larger classes of sigmoid activation feedforward neural networks can be implemented in polynomial size Boolean circuits with a small constant fan-in at the expense of a logarithmic factor increase in the number of layers. Similar results are obtained for threshold circuits, and are liked with the previous ones. The main conclusion is that there are interesting fan-in dependent depth-size tradeoffs when trying to digitally implement sigmoid activation feedforward neural networks. Copyright © 1996 Elsevier Science Ltd

The computational efficacy of finite-field arithmetic

We investigate the computational power of finite-field arithmetic operations as compared to Boolean operations. We pursue this goal in a representation-independent fashion. We define a good representation of the finite fields to be essentially one in which the field arithmetic operations have polynomial-size Boolean circuits. We exhibit a function ƒ p on the prime fields with two properties: first, ƒ p has a polynomial-size Boolean circuit in any good representation, i.e. ƒ p is easy to compute with general operations; second, any function that has polynomial-size Boolean circuits in some good representation also has polynomial-size arithmetic circuits if and only if ƒ p has polynomial-size arithmetic circuits. Informally, ƒ p is the hardest function to compute with arithmetic that has small Boolean circuits. We reduce the function ƒ p to the pair of functions g p = ∑ k=1 p−1 x k k on the field F p , and m p on Z p 2 . Here m p is the “modulo p” function defined in the natural way. We show that ƒ p has polynomial-size arithmetic circuits if and only if g p and m p have polynomial-size arithmetic circuits, the latter being arithmetic circuits over the ring Z p2 . Finally, we establish a connection of ƒ p and m p with the Bernoulli polynomials and determine the coefficients of the unique degree p − 1 polynomial over F p that computes ƒ p .

Factoring Rational Polynomials over the Complex Numbers

NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables (d, c, and n, respectively). If n is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor. After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of $(P(z_1 ,z_2 ) = 0)$ correspond to the connected components of the real surface (or complex curve) $P(z_1 ,z_2 ) = 0$ minus its singular points. In finding the number of connected components of the surface $P = 0$, the surface is projected to the the $z_2 $-plane. The singular points of $P(z_1 ,z_2 )$ lie over the projection’s critical values. The inverse image of a grid isolating the critical values in the $z_2 $-plane lifts to a one-dimensional real curve skeleton on the surface $(P = 0)$ whose number of connected components is precisely the number of connected components of $P = 0$ minus its singular points. The connectivity of this curve skeleton is constructed symbolically using Sturm sequences associated with the various polynomials defining these maps. Given the number of irreducible factors and their degrees, the actual factors can be reconstructed using the recent result of Neff [Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 152–162] on finding zeros of one-variable polynomials in NC.

Upper bounds on recognition of a hierarchy of non-context-free languages

Control grammars, a generalization of context-free grammars recently introduced for use in natural language recognition, are investigated. In particular, it is shown that a hierarchy of non-context-free languages, called control language hierarchy (CLH), generated by control grammars can be recognized in polynomial time. Previously, the best-known upper bound was exponential time. It is also shown that CLH is in NC (2), the class of languages recognizable by uniform boolean circuits of polynomial size and O(log 2 n) depth.

EFFICIENT DETECTORS AND CONSTRUCTORS FOR SIMPLE LANGUAGES

In this paper, we investigate the complexity of computing the detector, constructor and lexicographic constructor functions for a given language. The following classes of languages will be considered: (1) context-free languages, (2) regular sets, (3) languages accepted by one-way nondeterministic auxiliary pushdown automata, (4) languages accepted by one-way nondeterministic logspace-bounded Turing machines, (5) two-way deterministic pushdown automaton languages, (6) languages accepted by uniform families of constant-depth polynomial-size Boolean circuits, and (7) languages accepted by multihead finite automata. We show that for the classes (1)–(4), efficient detectors, constructors and lexicographic constructors exist, whereas for (5)– (7) polynomial-time computable detectors, constructors and lexicographic constructors exist iff there are no sparse sets in NP−P (or equivalently, E=NE). Our results provide sharp boundaries between classes of languages which have efficient detectors, constructors and classes of languages for which efficient detectors and constructors do not appear to exist.

Ranking and formal power series

We prove that the ranking problem for unambiguous context-free languages is NC 1-reducible to the value problem for algebraic formal power series in noncommuting variables. In the particular case of regular languages we show that the problem of ranking is NC 1-reducible to the problem of counting the number of strings of given length in suitable regular languages. As a consequence ranking problems for regular languages are NC 1-reducible to integer division and hence computable by log-space uniform boolean circuits of polynomial size and depth O(log n log log n), or by P-uniform boolean circuits of polynomial size and depth O(log n).

Very Fast Parallel Polynomial Arithmetic

Parallel algorithms for polynomial arithmetic—multiplication of n polynomials of degree m, polynomial division with remainder, and polynomial interpolation—are presented. These algorithms can be implemented using polynomial time constructible families of Boolean circuits of polynomial size and optimal order depth, or log space constructible families of polynomial size and near optimal depth, for computations over $\mathbb{Z},\mathbb{Q}$, finite fields, and several other domains. Arithmetic circuits of polynomial size and optimal order depth are obtained for polynomial arithmetic over arbitrary fields.

Fast Parallel Computation of Hermite and Smith Forms of Polynomial Matrices

Boolean circuits of polynomial size and polylogarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficient sequential algorithms. Furthermore, we give a polynomial-time deterministic sequential algorithm for the Smith normal form over the rationals. The Smith normal form algorithms are applied to the rational canonical form of matrices over finite fields and the field of rational numbers.

Fast parallel absolute irreducibility testing

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomial-time problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coeffieients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algorithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolute irreducible integral polynomial modulo p , the polynomial's irreducibility in the algebraic closure of the finite field of order p is not preserved.